Two possible solutions:
1) allow moderators to pre-select which forums get fed into the New Posts feed
2) allow the user to select what threads or forums they want to receive New Posts from.
It may be possible to free up some space in the submenu, by removing the "Search Forums" button, when you have the blank search bar right there next to it.
Once space is freed up, you could have two New Posts buttons:
1) "All New Posts" (unfiltered feeding from the whole site, as currently)
2) "Filtered New Posts" (feeding only out of the preselected sources above)